html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 EllipticF

 http://functions.wolfram.com/08.05.20.0005.02

 Input Form

 D[EllipticF[z, m], {z, n}] == KroneckerDelta[n] EllipticF[z, m] + ((I^(n - 1) 2^(n - 2))/(Sqrt[(E^(2 I z) m)/(m - 2 (1 + Sqrt[1 - m]))] Sqrt[1 - m Sin[z]^2])) Sqrt[(2 Sqrt[1 - m] + (1 - E^(2 I z)) m - 2)/ Sqrt[1 - m]] Sum[((E^(2 I k z) StirlingS2[n - 1, k] Pochhammer[1/2, k] (-m)^k)/(2 + 2 Sqrt[1 - m] + (E^(2 I z) - 1) m)^k) AppellF1[1/2, -(1/2), 1/2, 1/2 - k, (2 + 2 Sqrt[1 - m] + (E^(2 I z) - 1) m)/(2 + 2 Sqrt[1 - m] - m), (2 + 2 Sqrt[1 - m] + (E^(2 I z) - 1) m)/(4 Sqrt[1 - m])], {k, 0, n - 1}] /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["n", "-", "2"]]]]], RowBox[List[SqrtBox[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", "m"]], RowBox[List["m", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "m"]]]]], ")"]]]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]]], ")"]], " ", "m"]], "-", "2"]], SqrtBox[RowBox[List["1", "-", "m"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "z"]]], RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], ")"]], "k"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", FractionBox[RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "m"]]], ",", FractionBox[RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n F ( z m ) z n δ KroneckerDelta n F ( z m ) + n - 1 2 n - 2 2 z m m - 2 ( 1 - m + 1 ) 1 - m sin 2 ( z ) ( 1 - 2 z ) m + 2 1 - m - 2 1 - m k = 0 n - 1 2 k z 𝒮 TagBox["\[ScriptCapitalS]", StirlingS2] n - 1 ( k ) ( - m ) k ( ( 2 z - 1 ) m + 2 1 - m + 2 ) k ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] F 1 AppellF1 ( 1 2 ; - 1 2 , 1 2 ; 1 2 - k ; ( 2 z - 1 ) m + 2 1 - m + 2 - m + 2 1 - m + 2 , ( 2 z - 1 ) m + 2 1 - m + 2 4 1 - m ) /; n Condition z n EllipticF z m KroneckerDelta n EllipticF z m n -1 2 n -2 2 z m m -1 2 1 -1 m 1 2 1 -1 1 2 1 -1 m z 2 1 2 -1 1 -1 2 z m 2 1 -1 m 1 2 -2 1 -1 m 1 2 -1 1 2 k 0 n -1 2 k z StirlingS2 n -1 k -1 m k 2 z -1 m 2 1 -1 m 1 2 2 k -1 Pochhammer 1 2 k AppellF1 1 2 -1 1 2 1 2 1 2 -1 k 2 z -1 m 2 1 -1 m 1 2 2 -1 m 2 1 -1 m 1 2 2 -1 2 z -1 m 2 1 -1 m 1 2 2 4 1 -1 m 1 2 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["KroneckerDelta", "[", "n", "]"]], " ", RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "1"]]], " ", SuperscriptBox["2", RowBox[List["n", "-", "2"]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]]]], ")"]], " ", "m"]], "-", "2"]], SqrtBox[RowBox[List["1", "-", "m"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "z"]]], " ", RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], "k"]]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", "k"]], ",", FractionBox[RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "-", "m"]]], ",", FractionBox[RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["2", "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "m"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], "-", "1"]], ")"]], " ", "m"]]]], ")"]], "k"]]]]]], RowBox[List[SqrtBox[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", "m"]], RowBox[List["m", "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "m"]]]]], ")"]]]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2001-10-29